$x,y\in L$ tow algebraic element such that $xy,x+y\in K$. $x,y\in K$?

Question

if $L/K$ be a extension field and $x,y\in L$ tow algebraic element over K such that $xy,x+y\in K$. Can we say that $x,y\in K$?

Answer 1

No. Let $K = \mathbf Q$, $L= \mathbf Q[\sqrt 2]$, $x = -y = \sqrt 2$.